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Properties of Breuil-Kisin modules inherited by $p$-divisible groups

Mabud Ali Sarkar, Absos Ali Shaikh

Abstract

In this paper, by assuming a faithful action of a finite flat $\mathbb{Z}_p$-algebra $\mathscr{R}$ on a $p$-divisible group $\mathcal{G}$ defined over the ring of $p$-adic integers $\mathscr{O}_K$, we construct a category of new Breuil-Kisin module $\mathfrak{M}$ defined over the ring $\mathfrak{S}:=W(κ)[\![u]\!]$ and study the freeness and projectiveness properties of such a module.

Properties of Breuil-Kisin modules inherited by $p$-divisible groups

Abstract

In this paper, by assuming a faithful action of a finite flat -algebra on a -divisible group defined over the ring of -adic integers , we construct a category of new Breuil-Kisin module defined over the ring and study the freeness and projectiveness properties of such a module.

Paper Structure

This paper contains 3 sections, 14 theorems, 14 equations.

Table of Contents

  1. Introduction and motivation
  2. Projectiveness and freeness of the objects in $\text{Mod}_{/\mathfrak{S}_\mathscr{R}}^{\varphi}$
  3. Main Results

Key Result

Theorem 2.1

If the finite flat $\mathbb{Z}_p$-algebra $\mathscr{R}$ contains all the embeddings $\sigma$ of $W(\kappa)$, then $\mathfrak{M}$ is free $(\mathscr{R} \otimes_{\mathbb{Z}_p}W(\kappa))[\![u]\!]$-module.

Theorems & Definitions (29)

  • Definition 2.1
  • Remark 1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • ...and 19 more